Mindlin’s Micro-structural and Gradient Elasticity Theories and Their Thermodynamics

Particular gradient elasticity models arise as limits of Mindlin’s micro-structured elastic materials, when the micro-deformation is assumed to be a function of the classical displacement gradient. In this case, the only independent kinematical degrees of freedom in the internal points of the continuum are the classical ones, reflected by the displacement field. The present work addresses a class of gradient elasticity models, including those of Mindlin, in the context of a continuum theory without balance laws of double-forces and without parts of stresses being left indeterminate. This is accomplished by using a non-conventional thermodynamic framework. The resulting models are restated as classical continua with non-classical body and non-classical inertial forces. The new interpretation suggests a definition of the appropriate boundary conditions in a different way than usually done in references addressing this subject. Numerical examples involving the axial motion of a gradient elastic bar possessing micro-stiffness and micro-inertia are studied in detail to illustrate the behavior of the above models.